I have a very similar problem with WordPress and Litespeed Web Server Enterprise v4.1.13:
I'm also using WP-Super-Cache that creates index.html.gz files for every web page. Sometimes I can see the website without any problems, and sometimes the browser shows me the gz-file. I used the web-sniffer.net service to to find out what exactly happens and causes the strange behavior.
I have figured out that sometimes LiteSpeed sends "Content-Encoding: gzip" with the HTTP header, and sometimes it does not. When it does not send this, the browser doesn't know that it has to decompress the gzip-content and shows the plain gzip-file. So I suspect that LiteSpeed does not always recognize the gzip-content and therefore it does not add the content-encoding to the HTTP header.
It does not matter if I enable or disable gzip-compression in LiteSpeed configuration. It's just that sometimes LiteSpeed delivers the gz-file from WP-Cache with "Content-Encoding: gzip" information, and sometimes it sends the gz-file without this important piece of information.
Any ideas what could be the reason for this problem?